Question: If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $43^\circ,$ what is the angle between the vectors $-\mathbf{a}$ and $\mathbf{b}$?
Since $\mathbf{a}$ and $-\mathbf{a}$ point in opposite directions, the angle between them is $180^\circ.$  Then the angle between $-\mathbf{a}$ and $\mathbf{b}$ is $180^\circ - 43^\circ = \boxed{137^\circ}.$

[asy]
unitsize(2 cm);

pair A, B, O;

A = 2*dir(12);
B = dir(12 + 43);
O = (0,0);

draw(O--A,red,Arrow(6));
draw(O--B,red,Arrow(6));
draw(O--(-A),red,Arrow(6));

label("$\mathbf{a}$", (O + A)/2, S);
label("$\mathbf{b}$", (O + B)/2, NW);
label("$-\mathbf{a}$", (O + (-A))/2, S);
label("$43^\circ$", (0.4,0.25));
label("$137^\circ$", (-0.15,0.15));
[/asy]